The Asymptotic Existence of Resolvable Group Divisible Designs
Bibliographic record
Abstract
Abstract A group divisible design (GDD) is a triple which satisfies the following properties: (1) is a partition of X into subsets called groups; (2) is a collection of subsets of X, called blocks, such that a group and a block contain at most one element in common; and (3) every pair of elements from distinct groups occurs in a constant number λ blocks. This parameter λ is usually called the index. A k‐GDD of type is a GDD with block size k, index , and u groups of size g. A GDD is resolvable if the blocks can be partitioned into classes such that each point occurs in precisely one block of each class. We denote such a design as an RGDD. For fixed integers and , we show that the necessary conditions for the existence of a k‐RGDD of type are sufficient for all . As a corollary of this result and the existence of large resolvable graph decompositions, we establish the asymptotic existence of resolvable graph GDDs, G‐RGDDs, whenever the necessary conditions for the existence of ‐RGDs are met. We also show that, with a few easy modifications, the techniques extend to general index. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 112–126, 2013
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.016 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".